For instance, note that a `quantum formalism' can be defined also over the real instead of the complex numbers (see e.g. [1]}, [2]}). More generally, several results aiming to reconstruct quantum mechanics arrive at the level of (special) Jordan algebras corresponding to associative algebras over the real, complex and quaterionic numbers (e.g. [3]}, [4]}, [5]}, [6]}). In this context, it is interesting to note that time orientations define a complex structure on (the order derivations of) \(\mathcal {J}(\mathcal {A})\)  [7]}, [8]}. Compare this with Eq. (REF ), where we used the complex structure of the associative algebra \(\mathcal {A}\)  implicitly to define the canonical time orientation \(\Psi _\mathcal {A}\) . In this way, dynamical correspondences can be seen as a justification for the prominence of complex numbers in quantum mechanics [9]}.For the intimate relationship between dynamical correspondences and Noether's theorem, see [10]}. By Thm. REF , these arguments are further inherently connected with quantum entanglement.
